Download Question and Problem Answers Chapter 5

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Question and Problem Answers
Chapter 5 - Modern Portfolio Theory
2
5 - 1:
Shares
100,000 Alliance Gaming
25,000 Burlington Northern
25,000 Ameren Corp
Cash
Price
4.000
35.000
25.000
Market Value
$400,000.
$875,000.
$625,000.
$100,000.
$2,000,000.
TOTAL PORTFOLIO
2
â
2.00
0.92
0.71
0.00
$400,000 * 2.00 =
$875,000 * 0.92 =
$625,000 * 0.71 =
$100,000 * 0.00 =
$2,048,750. / $2,000,000. =
800,000.00
805,000.00
443,750.00
0.00
2,048,750.00
1.024375
A.
Cash has a â = 0.00. The return on cash, if any, is independent of conditions in the equity market.
B.
â = 1.024
C.
The portfolio â characterizes this portfolio as fairly close to the market. On average we would expect this portfolio
to move with the market. We expect that when the market declines by 1% this portfolio would decline by 1.024%, but
if the market advances 1% this portfolio would advance by 1.024%.
5 - 2:
The first thing to do is to write down the information you are given: The weighted average of the 20 stocks in the portfolio
is 0.60. One of the twenty stocks has a â = 0.35. We do not know any of the other âs but we do know that each of the
twenty stocks has a market value of $100,000. Thus
Simple algebra gives us
and we can put Caterpillar's â of 1.12 into the portfolio â calculation.
We could accomplish the same thing by calculating:
2
2
FINANCIAL MARKETS ... AND THE INSTRUMENTS THAT TRADE IN THEM
5 - 3:
The SML shows us how much investors require in compensation for the systematic risk they bear. Investors require some
return for postponing consumption. This return is the intercept of the SML and represents investments with no systematic
risk. In other words, all investments must earn at least this minimum return, otherwise investors will consume their income
rather than investing it.
Stocks with an "average" amount of systematic risk (â=1; the market risk) should provide a return equal to the average of
all investments. Thus the SML must pass through both the risk free rate of return (â=0) and the market rate of return (â=1).
The SML has a positive slope. This reflects the premise that investors are risk averse: the more risky an investment is, the
greater the return an investor will demand for investing in it. The slope of the line is known as the risk premium; this is the
premium that investors demand by taking on an additional unit of systematic risk as measured by â.
Note that the SML and CAPM deal solely in systematic risk. Unsystematic risk is assumed eliminated through
diversification.
@ Finance 254 [G. McIntire]
2
5 - 4:
A.
The T-bill return does not depend on the state of the economy because the Treasury must and will redeem the bills
at par regardless. The T-Bills are risk free in the nominal sense because the 8% return will be realized in all possible
states. However, this 8% return is composed of the real risk free rate (say 3%) plus an inflation premium (5%). If
inflation averaged 9% over the year rather than the expected 5% then the real returns realized on the T-bills would
be -1%. Thus, in real terms, T-bills are not riskless.
B.
Compulectric's returns are positively correlated with the economy because the firm's sales, and hence profits, will
generally experience the same ups and downs as the economy as a whole. If the economy grows, so will
Compulectrics. On the other hand Gold Mines is considered a hedge against both slow growth and high inflation.
When the market turns sour investors will retreat into gold and gold production. This stock is thus counter-cyclical.
(This is a totally fictional stock. It is almost impossible to find real stocks that run counter to the market on a
systematic basis. Even Homestake Mining had a low but positive â (.35)).
E[R]
T-Bills
Index Fund
Compulectrics
American Rubber
Gold Mines
To demonstrate, calculate for the Index Fund
8.0%
15.0%
17.4%
13.8%
1.7%
ó
0.0
15.3
20.0
18.8
13.4
CV =
ó
E[R]
0.00
1.02
1.15
1.36
7.88
CHAPTER 5 - MODERN PORTFOLIO THEORY
3
C.
If the trust is risk neutral we would invest in Compulectrics; it has the highest expected return.
D.
American Rubber has a higher ó than the market fund yet is expected to earn a lower return. When we measure risk
per unit of expected return, Gold Mines, with its low expected rate of return, becomes the most risky stock.
E.
We calculate the covariance as follows:
Scenario
Compulectrics:
R - E[RC]
Gold Mines:
R - E[RGM]
(R - E[RC]) (R - E[RGM])
Rapid Growth
50.0 - 17.4 = 32.6%
-20.0 - 1.74 = -21.74%
-708.724
Above Average
35.0 - 17.4 = 17.6%
-10.0 - 1.74 = -11.74%
-206.624
Average
20.0 - 17.4 =
0.0 - 1.74 = -1.74%
-4.524
Below Average
-2.0 - 17.4 = -19.4%
14.7 - 1.74 = 12.96%
-251.424
-22.0 - 17.4 = -39.4%
28.0 - 1.74 = 26.26%
-1034.644
Recession
2.6%
Covariance:
-267.756
The covariance is -267.756% and the correlation coefficient is
There are two ways to calculate the standard deviation of the portfolio.
One, you can use the correlation coefficient you just calculated:
Two, you can calculate the portfolio's return in each state of the
economy and then calculate
E[R] =
E[R2 ] =
ó=
CV =
9.57%
102.7145%,
3.3, and
0.34.
The portfolio risk is much lower than the risk associated with either
of the individual stocks. This is because the two stocks are negatively
correlated: when Compulectrics goes down Gold Mines goes up, and
vice versa.
State of Economy
Prob
E[R]
Rapid Growth
0.1
15.00%
Above Average
0.2
12.50%
Average
0.4
10.00%
Below Average
0.2
6.35%
Recession
0.1
3.00%
4
FINANCIAL MARKETS ... AND THE INSTRUMENTS THAT TRADE IN THEM
F.
Use the T-bill rate to proxy the risk free rate of return. The expected market rate of return is estimated at 15.0%
(remember the index fund), thus the risk premium is 7.0% (15.0 - 8.0). The rates of return required to justify the risk
are:
CAPM: Required Return
E[R]
Compulectrics
8.0 + 1.29 (7.0) = 17.0%
17.4% undervalued (buy)
American Rubber
8.0 + 0.68 (7.0) = 12.8%
13.8% undervalued (buy)
Gold Mines
8.0 - 0.86 (7.0) = 2.0%
1.7%
overvalued (sell)
Portfolio
â = 0.215 [ â=.5(1.29) + .5(-0.86)]
8.0 + .215 (7.0) = 9.505%
9.6%
slightly undervalued
G.
The undervalued securities are above the line; their rate of return is higher than the return required to compensate for
the risk. The overvalued securities are below the line; their rate of return is less than the return required to compensate
for the risk.
H.
If inflation expectations rise by 3% then the risk free rate increases from 8% to 11% and the market rate increases from
15% to 18%, generating the equation R = 11.0 + â (7.0). All required returns increase by 3 percentage points. Note
that the expected returns will also change. Run back to the investment guru and have his department regenerate the
expected returns on these stocks. Note that the market risk premium remains at 7%.
I.
When investors' risk aversion increases, the market risk premium increases, in this case from 7% (15.0 - 8.0 = 7%)
to 10% (18.0 - 8.0 = 10%). The required rate of return will increase sharply on high risk (high â) stocks, but will
increase only marginally on low risk stocks.
CHAPTER 5 - MODERN PORTFOLIO THEORY
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J.
CAPM: required Return
E[R]
Compulectrics
8.0 + 1.29 (10.0) = 20.9%
17.4
overvalued (sell)
American Rubber
8.0 + 0.68 (10.0) = 14.8%
13.8%
overvalued (sell)
Gold Mines
8.0 - 0.86 (10.0) = -0.6%
1.7%
undervalued (buy)
Portfolio
â = 0.215 [â=.5(1.29) + .5(-0.86)]
8.0 + .215 (10.0) = 10.15%
9.6%
overvalued (sell)
Gold Mines has a negative required return. Investing in a stock that has a negative required return is just like buying
an insurance policy; insurance also has a negative expected return. A negative beta stock is insurance against a bear
market.
Note how American Rubber and Compulectrics changes from a buy to a sell and Gold Mines changes from a sell to
a buy when the risk premium changes from 7% to 10%.
2
5 - 5:
A.
E[R] = 3.3% + â [ 12.3% - 3.3%] = 3.3% + â 9.0%
B.
Intercept = 3.3% and slope = 9.0%
C.
E[R] = 3.3% + (0.8) 9.0% = 10.5%
D.
Abnormal Return = Actual Return - Normal Return = 14.8% - 10.5% = +4.3%
(Graphed below)
6
FINANCIAL MARKETS ... AND THE INSTRUMENTS THAT TRADE IN THEM
E.
We calculate the Sharpe and Treynor measures as follows:
T-Bills
Market Index
Fund
Discovery Café
The Treynor measure for the equity index will always be equal to the risk premium whereas the Sharpe measure will capture
the standard deviation of the market itself.
If the Treynor measure for Discovery Café is equal to the risk premium of 9.0% then Discovery Café adds the same risk
and return to the portfolio as does the market index itself. If the Treynor measure is greater than the risk premium then
Discovery Café adds more return than risk; if less than the risk premium then it adds more risk than return.
To see this calculate what the return on Discovery Café must be to get a Treynor measure equal to 9.0%. What must it be
to get a measure higher than 9.0%? Lower then 9.0%?
CHAPTER 5 - MODERN PORTFOLIO THEORY
2
7
5 - 6:
A.
Remember to round the market value in each month to two decimal places before continuing.
S&P 500
IBM
GM
$10,000.00
B.
$10,000.00
2001
-9.9%
-1.4%
$9,860.00
-2.6%
$9,740.00
2002
11.3%
13.9%
$11,230.54
31.2%
$12,778.88
2003
7.5%
34.3%
$15,082.62
29.7%
$16,574.21
2004
12.9%
-14.4%
$12,910.72
16.9%
$19,375.25
2005
-1.3%
2.5%
$13,233.49
-38.4%
$11,935.15
Use the [data] and [stat] in your business calculator to calculate the statistics on IBM. Enter the observations on IBM
into X and the observations on the index into Y.
IBM
1. E[R]= 6.98%
2. ó= 16.38%
3. s = 18.31
4. á = 3.60
5. â = 0.07
C.
Use [data] to enter the GM stats into X.
GM
1.
2.
3.
4.
5.
D.
E[R] = 7.36%
ó = 25.89%
s = 28.95%
á = 2.52
â = 0.21
Use [data] to enter the IBM stats into Y. You can double check your data entry by verifying the answers to 1, 2, and
3 in questions B. and C.
ñ = 0.405 (40.5%)
2
5 - 7:
Factor
Security
IBM
Compaq
AT&T
Motorola
Ford
(IBM)
(CPQ)
(T)
(MOT)
(F)
Risk Free rate
F1
1.0
1.0
1.0
1.0
1.0
Market Index
F2
1.1
0.9
0.2
0.8
0.2
GDP
F3
0.5
0.4
0.5
0.5
0.6
Expected Return
22.0%
18.8%
13.0%
12.0%
14.2%
8
FINANCIAL MARKETS ... AND THE INSTRUMENTS THAT TRADE IN THEM
A.
F1 = 5; F2 = 10; F3 = 12
For each stock R = â1 F1 + â2 F2 + â3 F3. Using the information given in the question we can generate three equations
in three unknowns:
IBM
Compaq
AT&T
1.0 F1
1.0 F1
1.0F1
+ 1.1 F2
+ 0.9 F2
+ 0.2 F2
+ 0.5 F3 =
+ 0.4 F3 =
+ 0.5 F3 =
22.0%
18.8%
13.0%
There are several ways to solve the system of equations. One simple way is
1) subtract AT&T from IBM
IBM
AT&T
F1
- F1
+ 1.1 F2
- 0.2 F2
0.9 F2
F2
+ 0.5 F3 =
- 0.5 F3 =
=
=
22.0
- 13.0
9.0
10.0
2) substitute for F2 and subtract Compaq from IBM
IBM
Compaq
F1
- F1
+ 0.5 F3
- 0.4 F3
.1 F3
F3
=
=
=
=
11.0
- 9.8
1.2
12.0
3) substitute in any equation to get F1 = 5.0
B.
Motorola:
1.0 (5.0) + 0.8 (10.0) + 0.5 (12.0) = 19.0%
Motorola is overpriced because we would require a return of 19.0% to compensate for the risk exposure but, at the
current price, we expect to generate only a 12.0% return.
C.
Put X% in IBM, Y% in Compaq, and Z% in AT&T. The resulting portfolio will then have a beta for each factor equal
to the weighted average beta for that factor. The weights are then adjusted so that the portfolio beta equals Motorola’s
beta for each factor.
Risk Free Rate (F1)
Market Index (F2)
GDP (F3)
IBM Compaq
1.0 X + 1.0 Y
1.1 X + 0.9 Y
0.5 X + 0.4 Y
AT&T
+ 1.0 Z =
+ 0.2 Z =
+ 0.5 Z =
1.0
0.8
0.5
1) subtract 2*F3 from F1
(F1)
(2F3)
1.0 X + 1.0 Y
- 1.0 X - 0.8 Y
0.2 Y
Y
+ 1.0 Z = 1.0
- 1.0 Z = -1.0
= 0.0
= 0.0
1.0 X
- 5.5 X
- 4.5 X
X
+1.0 Z =
- 1.0 Z =
=
=
2) substitute for Y and subtract 5*F2 from F1
(F1)
(5F2)
3) substitute in F1 to get Z = 1/3
1.0
- 4.0
- 3.0
2
/3
CHAPTER 5 - MODERN PORTFOLIO THEORY
9
Construct a portfolio of 2 /3 IBM and 1 /3 AT&T. The portfolio’s betas will be 1.0, 0.8, and 0.5 (identical to Motorola’s)
but the expected return is 19.0%.
The portfolio at an expected return of 19.0% is a better investment than Motorola at 12.0% since both investments
have the same market risk.
D.
Ford: 1.0 (5.0) + 0.2 (10.0) + 0.6 (12.0) = 14.2%
Ford is correctly priced.
E.
Put X% in IBM, Y% in Compaq, and Z% in AT&T. The resulting portfolio will then have a beta for each factor equal
to the weighted average beta for that factor. The weights are then adjusted so that the portfolio beta equals Ford’s beta
for each factor.
Risk Free Rate (F1)
Market Index (F2)
GDP (F3)
1.0 X + 1.0 Y + 1.0 Z = 1.0
1.1 X + 0.9 Y + 0.2 Z = 0.2
0.5 X + 0.4 Y + 0.5 Z = 0.6
1) subtract 2*F3 from F1
(F1)
(2F3)
1.0 X + 1.0 Y
- 1.0 X - 0.8 Y
0.2 Y
Y
+ 1.0 Z =
- 1.0 Z =
=
=
1.0
-1.2
-0.2
-1.0
1.0 X
- 5.5 X
- 4.5 X
X
+1.0 Z =
- 1.0 Z =
=
=
2.0
- 5.5
- 3.5
7
/9
2) substitute for Y and subtract 5*F2 from F1
(F1)
(5F2)
3) substitute in F1 to get Z = 11/9
Construct a portfolio of 7 /9 IBM, 11/9 AT&T and short sell Compaq. The portfolio’s betas will also be 1.0, 0.8, and
0.5 and the expected return is also 14.2%.
The portfolio and Ford have the same betas and the same expected return. Ford is priced correctly.
©2006
ELISABETH OLTHETEN AND KEVIN G. WASPI
Not to be transmitted, copied, or distributed without express written permission from the authors.